<?xml version="1.0" encoding="ISO-8859-1"?>

<!DOCTYPE proofs SYSTEM "proof3.dtd">

<?xml-stylesheet type="text/xsl" href="proof3.xsl"?>

<proofs number_of_proofs="1">

<source>pair.out</source>

<heading><![CDATA[
Prover9 (32) version August-2006A, August 2006.
Process 11624 was started by mccune on cleo.thornwood,
Sat Aug 12 21:15:08 2006
The command was "/home/mccune/bin/prover9 -f pair.in".
]]></heading>

<proof number="1" length="136" max_count="43">

<comments><![CDATA[
% Proof 1 at 419.45 (+ 2.48) seconds: "Sheffer".
% Length of proof is 136.
% Level of proof is 18.
% Maximum clause weight is 43.
% Given clauses 1404.
]]></comments>

  <clause id="1">
    <literal><![CDATA[
      f(x,y) = f(y,x)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="2">
    <literal><![CDATA[
      f(f(x,y),f(x,f(y,z))) = x
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="3">
    <literal><![CDATA[
      f(f(a,a),f(a,a)) != a
    ]]></literal>
    <literal><![CDATA[
      f(a,f(b,f(b,b))) != f(a,a)
    ]]></literal>
    <literal><![CDATA[
      f(f(f(b,b),a),f(f(c,c),a)) != f(f(a,f(b,c)),f(a,f(b,c)))
    ]]></literal>
    <attribute><![CDATA[
      answer("Sheffer")
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="4">
    <literal><![CDATA[
      f(f(a,a),f(a,a)) != a
    ]]></literal>
    <literal><![CDATA[
      f(a,f(b,f(b,b))) != f(a,a)
    ]]></literal>
    <literal><![CDATA[
      f(f(a,f(b,c)),f(a,f(b,c))) != f(f(a,f(b,b)),f(a,f(c,c)))
    ]]></literal>
    <attribute><![CDATA[
      answer("Sheffer")
    ]]></attribute>
    <justification jstring="[copy(3),rewrite(1(25),1(30)),flip(c)].">
      <j1 rule="copy" parents="3"/>
      <j2 rule="rewrite" parents="1 1"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="5">
    <literal><![CDATA[
      f(f(x,y),f(y,f(x,z))) = y
    ]]></literal>
    <justification jstring="[para(1(a,1),2(a,1,1))].">
      <j1 rule="para" parents="1 2"/>
    </justification>
  </clause>

  <clause id="6">
    <literal><![CDATA[
      f(f(x,y),f(x,f(z,y))) = x
    ]]></literal>
    <justification jstring="[para(1(a,1),2(a,1,2,2))].">
      <j1 rule="para" parents="1 2"/>
    </justification>
  </clause>

  <clause id="7">
    <literal><![CDATA[
      f(f(x,y),f(f(y,z),x)) = x
    ]]></literal>
    <justification jstring="[para(1(a,1),2(a,1,2))].">
      <j1 rule="para" parents="1 2"/>
    </justification>
  </clause>

  <clause id="8">
    <literal><![CDATA[
      f(x,f(f(x,y),f(f(x,f(y,z)),u))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(2(a,1),2(a,1,1))].">
      <j1 rule="para" parents="2 2"/>
    </justification>
  </clause>

  <clause id="9">
    <literal><![CDATA[
      f(x,f(x,f(x,y))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(2(a,1),2(a,1,2)),rewrite(1(2),1(3))].">
      <j1 rule="para" parents="2 2"/>
      <j2 rule="rewrite" parents="1 1"/>
    </justification>
  </clause>

  <clause id="10">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(y,x)) = x
    ]]></literal>
    <justification jstring="[para(5(a,1),1(a,1)),flip(a)].">
      <j1 rule="para" parents="5 1"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="11">
    <literal><![CDATA[
      f(f(x,y),f(y,f(z,x))) = y
    ]]></literal>
    <justification jstring="[para(1(a,1),5(a,1,2,2))].">
      <j1 rule="para" parents="1 5"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      f(f(x,y),f(f(x,z),y)) = y
    ]]></literal>
    <justification jstring="[para(1(a,1),5(a,1,2))].">
      <j1 rule="para" parents="1 5"/>
    </justification>
  </clause>

  <clause id="13">
    <literal><![CDATA[
      f(x,f(f(y,x),f(f(x,f(y,z)),u))) = f(y,x)
    ]]></literal>
    <justification jstring="[para(5(a,1),2(a,1,1))].">
      <j1 rule="para" parents="5 2"/>
    </justification>
  </clause>

  <clause id="15">
    <literal><![CDATA[
      f(x,f(f(x,f(y,z)),f(f(x,y),u))) = f(x,f(y,z))
    ]]></literal>
    <justification jstring="[para(2(a,1),5(a,1,1))].">
      <j1 rule="para" parents="2 5"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      f(f(f(x,y),z),f(z,x)) = z
    ]]></literal>
    <justification jstring="[para(2(a,1),5(a,1,2,2))].">
      <j1 rule="para" parents="2 5"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      f(x,f(f(x,f(y,z)),f(f(y,x),u))) = f(x,f(y,z))
    ]]></literal>
    <justification jstring="[para(5(a,1),5(a,1,1))].">
      <j1 rule="para" parents="5 5"/>
    </justification>
  </clause>

  <clause id="18">
    <literal><![CDATA[
      f(f(f(x,y),z),f(z,y)) = z
    ]]></literal>
    <justification jstring="[para(5(a,1),5(a,1,2,2))].">
      <j1 rule="para" parents="5 5"/>
    </justification>
  </clause>

  <clause id="22">
    <literal><![CDATA[
      f(x,f(f(x,y),f(y,z))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(2(a,1),6(a,1,2)),rewrite(1(4))].">
      <j1 rule="para" parents="2 6"/>
      <j2 rule="rewrite" parents="1"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      f(x,f(f(x,f(y,z)),f(f(x,z),u))) = f(x,f(y,z))
    ]]></literal>
    <justification jstring="[para(6(a,1),5(a,1,1))].">
      <j1 rule="para" parents="6 5"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      f(x,f(f(y,x),f(z,f(x,f(y,u))))) = f(y,x)
    ]]></literal>
    <justification jstring="[para(5(a,1),6(a,1,1))].">
      <j1 rule="para" parents="5 6"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      f(x,f(f(y,x),f(y,z))) = f(y,x)
    ]]></literal>
    <justification jstring="[para(5(a,1),6(a,1,2)),rewrite(1(4))].">
      <j1 rule="para" parents="5 6"/>
      <j2 rule="rewrite" parents="1"/>
    </justification>
  </clause>

  <clause id="27">
    <literal><![CDATA[
      f(x,f(f(x,y),f(z,y))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(6(a,1),6(a,1,2)),rewrite(1(4))].">
      <j1 rule="para" parents="6 6"/>
      <j2 rule="rewrite" parents="1"/>
    </justification>
  </clause>

  <clause id="31">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(z,x)) = x
    ]]></literal>
    <justification jstring="[para(5(a,1),7(a,1,2,1))].">
      <j1 rule="para" parents="5 7"/>
    </justification>
  </clause>

  <clause id="32">
    <literal><![CDATA[
      f(x,f(y,f(x,f(y,z)))) = f(x,f(y,z))
    ]]></literal>
    <justification jstring="[para(5(a,1),7(a,1,2)),rewrite(1(3),1(4))].">
      <j1 rule="para" parents="5 7"/>
      <j2 rule="rewrite" parents="1 1"/>
    </justification>
  </clause>

  <clause id="35">
    <literal><![CDATA[
      f(f(x,x),f(x,y)) = x
    ]]></literal>
    <justification jstring="[para(9(a,1),2(a,1,2))].">
      <j1 rule="para" parents="9 2"/>
    </justification>
  </clause>

  <clause id="38">
    <literal><![CDATA[
      f(a,f(b,f(b,b))) != f(a,a)
    ]]></literal>
    <literal><![CDATA[
      f(f(a,f(b,c)),f(a,f(b,c))) != f(f(a,f(b,b)),f(a,f(c,c)))
    ]]></literal>
    <attribute><![CDATA[
      answer("Sheffer")
    ]]></attribute>
    <justification jstring="[back_rewrite(4),rewrite(35(7)),xx(a)].">
      <j1 rule="back_rewrite" parents="4"/>
      <j2 rule="rewrite" parents="35"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="39">
    <literal><![CDATA[
      f(f(x,y),f(x,x)) = x
    ]]></literal>
    <justification jstring="[para(35(a,1),1(a,1)),flip(a)].">
      <j1 rule="para" parents="35 1"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="40">
    <literal><![CDATA[
      f(f(x,x),f(y,x)) = x
    ]]></literal>
    <justification jstring="[para(1(a,1),35(a,1,2))].">
      <j1 rule="para" parents="1 35"/>
    </justification>
  </clause>

  <clause id="44">
    <literal><![CDATA[
      f(x,f(f(x,x),y)) = f(x,x)
    ]]></literal>
    <justification jstring="[para(35(a,1),6(a,1,2)),rewrite(1(3))].">
      <j1 rule="para" parents="35 6"/>
      <j2 rule="rewrite" parents="1"/>
    </justification>
  </clause>

  <clause id="45">
    <literal><![CDATA[
      f(f(x,y),f(y,y)) = y
    ]]></literal>
    <justification jstring="[para(1(a,1),39(a,1,1))].">
      <j1 rule="para" parents="1 39"/>
    </justification>
  </clause>

  <clause id="46">
    <literal><![CDATA[
      f(x,f(f(x,y),f(z,f(x,x)))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(39(a,1),6(a,1,1))].">
      <j1 rule="para" parents="39 6"/>
    </justification>
  </clause>

  <clause id="54">
    <literal><![CDATA[
      f(f(f(x,y),z),f(x,f(f(f(x,y),z),f(f(f(x,y),f(z,u)),v)))) = f(f(f(x,y),z),f(f(f(x,y),f(z,u)),v))
    ]]></literal>
    <justification jstring="[para(8(a,1),7(a,1,2)),rewrite(1(8),1(11))].">
      <j1 rule="para" parents="8 7"/>
      <j2 rule="rewrite" parents="1 1"/>
    </justification>
  </clause>

  <clause id="61">
    <literal><![CDATA[
      f(x,f(y,f(x,x))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(40(a,1),5(a,1,2)),rewrite(1(3))].">
      <j1 rule="para" parents="40 5"/>
      <j2 rule="rewrite" parents="1"/>
    </justification>
  </clause>

  <clause id="63">
    <literal><![CDATA[
      f(x,f(f(y,x),f(z,f(x,x)))) = f(y,x)
    ]]></literal>
    <justification jstring="[para(45(a,1),6(a,1,1))].">
      <j1 rule="para" parents="45 6"/>
    </justification>
  </clause>

  <clause id="64">
    <literal><![CDATA[
      f(x,f(f(f(x,x),y),f(z,x))) = f(z,x)
    ]]></literal>
    <justification jstring="[para(45(a,1),7(a,1,1))].">
      <j1 rule="para" parents="45 7"/>
    </justification>
  </clause>

  <clause id="67">
    <literal><![CDATA[
      f(x,f(f(f(y,x),z),f(x,f(y,u)))) = f(x,f(y,u))
    ]]></literal>
    <justification jstring="[para(10(a,1),7(a,1,1))].">
      <j1 rule="para" parents="10 7"/>
    </justification>
  </clause>

  <clause id="70">
    <literal><![CDATA[
      f(f(x,y),f(f(z,x),y)) = y
    ]]></literal>
    <justification jstring="[para(1(a,1),11(a,1,2))].">
      <j1 rule="para" parents="1 11"/>
    </justification>
  </clause>

  <clause id="72">
    <literal><![CDATA[
      f(x,f(f(x,f(y,z)),f(u,f(x,y)))) = f(x,f(y,z))
    ]]></literal>
    <justification jstring="[para(2(a,1),11(a,1,1))].">
      <j1 rule="para" parents="2 11"/>
    </justification>
  </clause>

  <clause id="75">
    <literal><![CDATA[
      f(x,f(f(y,x),f(z,f(x,f(u,y))))) = f(y,x)
    ]]></literal>
    <justification jstring="[para(11(a,1),6(a,1,1))].">
      <j1 rule="para" parents="11 6"/>
    </justification>
  </clause>

  <clause id="79">
    <literal><![CDATA[
      f(x,f(y,f(x,f(z,y)))) = f(x,f(z,y))
    ]]></literal>
    <justification jstring="[para(11(a,1),7(a,1,2)),rewrite(1(3),1(4))].">
      <j1 rule="para" parents="11 7"/>
      <j2 rule="rewrite" parents="1 1"/>
    </justification>
  </clause>

  <clause id="86">
    <literal><![CDATA[
      f(x,f(f(f(x,y),z),f(x,f(y,u)))) = f(x,f(y,u))
    ]]></literal>
    <justification jstring="[para(2(a,1),12(a,1,1))].">
      <j1 rule="para" parents="2 12"/>
    </justification>
  </clause>

  <clause id="90">
    <literal><![CDATA[
      f(x,f(y,f(f(y,z),x))) = f(f(y,z),x)
    ]]></literal>
    <justification jstring="[para(12(a,1),7(a,1,2)),rewrite(1(3),1(4))].">
      <j1 rule="para" parents="12 7"/>
      <j2 rule="rewrite" parents="1 1"/>
    </justification>
  </clause>

  <clause id="96">
    <literal><![CDATA[
      f(x,f(f(f(y,x),z),f(x,f(u,y)))) = f(x,f(u,y))
    ]]></literal>
    <justification jstring="[para(11(a,1),12(a,1,1))].">
      <j1 rule="para" parents="11 12"/>
    </justification>
  </clause>

  <clause id="106">
    <literal><![CDATA[
      f(x,f(y,f(f(z,y),x))) = f(f(z,y),x)
    ]]></literal>
    <justification jstring="[para(18(a,1),6(a,1,2)),rewrite(1(3),1(4))].">
      <j1 rule="para" parents="18 6"/>
      <j2 rule="rewrite" parents="1 1"/>
    </justification>
  </clause>

  <clause id="142">
    <literal><![CDATA[
      f(x,f(f(y,f(z,x)),f(x,f(u,z)))) = f(x,f(u,z))
    ]]></literal>
    <justification jstring="[para(31(a,1),18(a,1,2)),rewrite(1(6))].">
      <j1 rule="para" parents="31 18"/>
      <j2 rule="rewrite" parents="1"/>
    </justification>
  </clause>

  <clause id="168">
    <literal><![CDATA[
      f(f(x,x),f(f(y,x),f(z,f(x,x)))) = f(z,f(x,x))
    ]]></literal>
    <justification jstring="[para(61(a,1),70(a,1,1))].">
      <j1 rule="para" parents="61 70"/>
    </justification>
  </clause>

  <clause id="172">
    <literal><![CDATA[
      f(f(x,f(f(x,y),z)),f(x,f(y,u))) = x
    ]]></literal>
    <justification jstring="[para(15(a,1),6(a,1,2))].">
      <j1 rule="para" parents="15 6"/>
    </justification>
  </clause>

  <clause id="200">
    <literal><![CDATA[
      f(x,f(y,f(x,y))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(31(a,1),15(a,1,2)),flip(a)].">
      <j1 rule="para" parents="31 15"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="205">
    <literal><![CDATA[
      f(x,f(y,f(y,x))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(1(a,1),200(a,1,2,2))].">
      <j1 rule="para" parents="1 200"/>
    </justification>
  </clause>

  <clause id="263">
    <literal><![CDATA[
      f(f(f(x,f(x,y)),z),f(f(y,y),z)) = z
    ]]></literal>
    <justification jstring="[para(205(a,1),70(a,1,2,1))].">
      <j1 rule="para" parents="205 70"/>
    </justification>
  </clause>

  <clause id="288">
    <literal><![CDATA[
      f(x,f(f(x,f(f(f(x,y),f(y,z)),u)),f(f(x,y),v))) = f(x,f(f(f(x,y),f(y,z)),u))
    ]]></literal>
    <justification jstring="[para(22(a,1),15(a,1,2,2,1))].">
      <j1 rule="para" parents="22 15"/>
    </justification>
  </clause>

  <clause id="295">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(y,f(f(x,f(y,z)),f(x,u)))) = f(f(x,f(y,z)),f(x,u))
    ]]></literal>
    <justification jstring="[para(25(a,1),7(a,1,2)),rewrite(1(5),1(8))].">
      <j1 rule="para" parents="25 7"/>
      <j2 rule="rewrite" parents="1 1"/>
    </justification>
  </clause>

  <clause id="314">
    <literal><![CDATA[
      f(f(x,f(f(y,x),z)),f(x,f(y,u))) = x
    ]]></literal>
    <justification jstring="[para(17(a,1),6(a,1,2))].">
      <j1 rule="para" parents="17 6"/>
    </justification>
  </clause>

  <clause id="354">
    <literal><![CDATA[
      f(f(x,y),f(y,f(z,f(y,f(u,x))))) = y
    ]]></literal>
    <justification jstring="[para(11(a,1),27(a,1,2,1)),rewrite(11(10))].">
      <j1 rule="para" parents="11 27"/>
      <j2 rule="rewrite" parents="11"/>
    </justification>
  </clause>

  <clause id="368">
    <literal><![CDATA[
      f(x,f(f(x,y),f(f(x,y),f(z,y)))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(27(a,1),200(a,1,2,2)),rewrite(1(5))].">
      <j1 rule="para" parents="27 200"/>
      <j2 rule="rewrite" parents="1"/>
    </justification>
  </clause>

  <clause id="568">
    <literal><![CDATA[
      f(x,f(f(x,f(y,z)),f(y,f(x,f(y,z))))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(32(a,1),200(a,1,2,2)),rewrite(1(6))].">
      <j1 rule="para" parents="32 200"/>
      <j2 rule="rewrite" parents="1"/>
    </justification>
  </clause>

  <clause id="569">
    <literal><![CDATA[
      f(x,f(y,y)) = f(x,f(y,x))
    ]]></literal>
    <justification jstring="[para(200(a,1),32(a,1,2))].">
      <j1 rule="para" parents="200 32"/>
    </justification>
  </clause>

  <clause id="581">
    <literal><![CDATA[
      f(x,f(f(x,f(y,z)),f(y,y))) = f(x,x)
    ]]></literal>
    <justification jstring="[back_rewrite(568),rewrite(569(6,R))].">
      <j1 rule="back_rewrite" parents="568"/>
      <j2 rule="rewrite" parents="569"/>
    </justification>
  </clause>

  <clause id="582">
    <literal><![CDATA[
      f(a,f(b,f(b,b))) != f(a,a)
    ]]></literal>
    <literal><![CDATA[
      f(f(a,f(b,c)),f(a,f(b,c))) != f(f(a,f(a,b)),f(a,f(a,c)))
    ]]></literal>
    <attribute><![CDATA[
      answer("Sheffer")
    ]]></attribute>
    <justification jstring="[back_rewrite(38),rewrite(569(27),1(26),569(32),1(31))].">
      <j1 rule="back_rewrite" parents="38"/>
      <j2 rule="rewrite" parents="569 1 569 1"/>
    </justification>
  </clause>

  <clause id="587">
    <literal><![CDATA[
      f(f(x,x),y) = f(y,f(x,y))
    ]]></literal>
    <justification jstring="[para(569(a,1),1(a,1)),flip(a)].">
      <j1 rule="para" parents="569 1"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="588">
    <literal><![CDATA[
      f(x,f(y,y)) = f(x,f(x,y))
    ]]></literal>
    <justification jstring="[para(1(a,1),569(a,2,2))].">
      <j1 rule="para" parents="1 569"/>
    </justification>
  </clause>

  <clause id="602">
    <literal><![CDATA[
      f(x,f(f(y,y),x)) = f(x,y)
    ]]></literal>
    <justification jstring="[para(35(a,1),569(a,1,2)),flip(a)].">
      <j1 rule="para" parents="35 569"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="603">
    <literal><![CDATA[
      f(f(x,x),f(f(x,y),f(x,y))) = f(x,f(x,x))
    ]]></literal>
    <justification jstring="[para(39(a,1),569(a,2,2)),rewrite(1(7))].">
      <j1 rule="para" parents="39 569"/>
      <j2 rule="rewrite" parents="1"/>
    </justification>
  </clause>

  <clause id="608">
    <literal><![CDATA[
      f(f(x,x),f(f(y,x),f(y,x))) = f(x,f(x,x))
    ]]></literal>
    <justification jstring="[para(45(a,1),569(a,2,2)),rewrite(1(7))].">
      <j1 rule="para" parents="45 569"/>
      <j2 rule="rewrite" parents="1"/>
    </justification>
  </clause>

  <clause id="644">
    <literal><![CDATA[
      f(f(f(x,y),z),f(f(y,f(x,x)),z)) = z
    ]]></literal>
    <justification jstring="[para(569(a,2),70(a,1,2,1))].">
      <j1 rule="para" parents="569 70"/>
    </justification>
  </clause>

  <clause id="740">
    <literal><![CDATA[
      f(f(x,x),y) = f(y,f(y,x))
    ]]></literal>
    <justification jstring="[para(1(a,1),587(a,2,2))].">
      <j1 rule="para" parents="1 587"/>
    </justification>
  </clause>

  <clause id="784">
    <literal><![CDATA[
      f(f(x,y),f(z,f(x,y))) = f(f(z,z),f(x,y))
    ]]></literal>
    <justification jstring="[para(587(a,1),15(a,2)),rewrite(15(9)),flip(a)].">
      <j1 rule="para" parents="587 15"/>
      <j2 rule="rewrite" parents="15"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="825">
    <literal><![CDATA[
      f(x,f(x,f(y,y))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(588(a,2),9(a,1,2))].">
      <j1 rule="para" parents="588 9"/>
    </justification>
  </clause>

  <clause id="915">
    <literal><![CDATA[
      f(f(f(f(x,x),y),z),f(f(y,x),z)) = z
    ]]></literal>
    <justification jstring="[para(602(a,1),70(a,1,2,1))].">
      <j1 rule="para" parents="602 70"/>
    </justification>
  </clause>

  <clause id="1950">
    <literal><![CDATA[
      f(x,f(y,f(x,f(f(x,y),z)))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(172(a,1),79(a,1,2)),flip(a)].">
      <j1 rule="para" parents="172 79"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="5676">
    <literal><![CDATA[
      f(x,f(y,f(x,f(f(y,x),z)))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(314(a,1),79(a,1,2)),flip(a)].">
      <j1 rule="para" parents="314 79"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="5859">
    <literal><![CDATA[
      f(f(x,y),f(f(x,y),f(x,y))) = f(x,f(x,x))
    ]]></literal>
    <justification jstring="[para(13(a,1),54(a,1,2,2)),rewrite(368(9),1(5),603(5),35(16),25(14)),flip(a)].">
      <j1 rule="para" parents="13 54"/>
      <j2 rule="rewrite" parents="368 1 603 35 25"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="7921">
    <literal><![CDATA[
      f(x,f(y,f(x,f(f(y,y),z)))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(35(a,1),581(a,1,2,2)),rewrite(1(4))].">
      <j1 rule="para" parents="35 581"/>
      <j2 rule="rewrite" parents="1"/>
    </justification>
  </clause>

  <clause id="7931">
    <literal><![CDATA[
      f(x,f(y,f(y,y))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(13(a,1),581(a,1,2,1)),rewrite(5859(5))].">
      <j1 rule="para" parents="13 581"/>
      <j2 rule="rewrite" parents="5859"/>
    </justification>
  </clause>

  <clause id="7997">
    <literal><![CDATA[
      f(f(a,f(b,c)),f(a,f(b,c))) != f(f(a,f(a,b)),f(a,f(a,c)))
    ]]></literal>
    <attribute><![CDATA[
      answer("Sheffer")
    ]]></attribute>
    <justification jstring="[back_unit_del(582),unit_del(a,7931)].">
      <j1 rule="back_unit_del" parents="582"/>
      <j2 rule="unit_del" parents="7931"/>
    </justification>
  </clause>

  <clause id="8021">
    <literal><![CDATA[
      f(f(x,f(x,x)),y) = f(y,y)
    ]]></literal>
    <justification jstring="[para(7931(a,1),1(a,1)),flip(a)].">
      <j1 rule="para" parents="7931 1"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="8030">
    <literal><![CDATA[
      f(f(x,x),f(y,f(y,y))) = x
    ]]></literal>
    <justification jstring="[para(7931(a,2),35(a,1))].">
      <j1 rule="para" parents="7931 35"/>
    </justification>
  </clause>

  <clause id="8056">
    <literal><![CDATA[
      f(x,f(f(y,x),f(z,f(z,z)))) = f(y,x)
    ]]></literal>
    <justification jstring="[para(7931(a,2),25(a,1,2))].">
      <j1 rule="para" parents="7931 25"/>
    </justification>
  </clause>

  <clause id="8061">
    <literal><![CDATA[
      f(x,f(y,f(z,f(z,z)))) = f(x,f(y,x))
    ]]></literal>
    <justification jstring="[para(7931(a,2),569(a,1,2))].">
      <j1 rule="para" parents="7931 569"/>
    </justification>
  </clause>

  <clause id="8063">
    <literal><![CDATA[
      f(f(x,f(x,x)),f(y,f(x,f(x,x)))) = y
    ]]></literal>
    <justification jstring="[para(7931(a,1),587(a,1)),rewrite(35(3)),flip(a)].">
      <j1 rule="para" parents="7931 587"/>
      <j2 rule="rewrite" parents="35"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="8065">
    <literal><![CDATA[
      f(f(x,f(x,x)),f(y,y)) = y
    ]]></literal>
    <justification jstring="[para(7931(a,1),587(a,2,2)),rewrite(8030(4)),flip(a)].">
      <j1 rule="para" parents="7931 587"/>
      <j2 rule="rewrite" parents="8030"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="8071">
    <literal><![CDATA[
      f(f(x,f(x,x)),f(f(x,f(x,x)),y)) = y
    ]]></literal>
    <justification jstring="[para(7931(a,1),740(a,1)),rewrite(35(3)),flip(a)].">
      <j1 rule="para" parents="7931 740"/>
      <j2 rule="rewrite" parents="35"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="8088">
    <literal><![CDATA[
      f(x,f(x,x)) = f(y,f(y,y))
    ]]></literal>
    <justification jstring="[para(7931(a,1),106(a,1,2)),rewrite(1(5),608(5),5859(7))].">
      <j1 rule="para" parents="7931 106"/>
      <j2 rule="rewrite" parents="1 608 5859"/>
    </justification>
  </clause>

  <clause id="8128">
    <literal><![CDATA[
      f(x,f(f(y,f(y,x)),f(z,f(z,z)))) = f(z,f(z,z))
    ]]></literal>
    <justification jstring="[para(7931(a,1),263(a,1,2)),rewrite(35(8),1(6))].">
      <j1 rule="para" parents="7931 263"/>
      <j2 rule="rewrite" parents="35 1"/>
    </justification>
  </clause>

  <clause id="8176">
    <literal><![CDATA[
      f(f(a,f(b,c)),f(x,f(x,x))) != f(f(a,f(a,b)),f(a,f(a,c)))
    ]]></literal>
    <attribute><![CDATA[
      answer("Sheffer")
    ]]></attribute>
    <justification jstring="[para(7931(a,2),7997(a,1))].">
      <j1 rule="para" parents="7931 7997"/>
    </justification>
  </clause>

  <clause id="8179">
    <literal><![CDATA[
      f(x,f(x,x)) = c_0
    ]]></literal>
    <justification jstring="[new_symbol(8088)].">
      <j1 rule="new_symbol" parents="8088"/>
    </justification>
  </clause>

  <clause id="8325">
    <literal><![CDATA[
      f(f(a,f(a,b)),f(a,f(a,c))) != f(c_0,f(a,f(b,c)))
    ]]></literal>
    <attribute><![CDATA[
      answer("Sheffer")
    ]]></attribute>
    <justification jstring="[back_rewrite(8176),rewrite(8179(7),1(7)),flip(a)].">
      <j1 rule="back_rewrite" parents="8176"/>
      <j2 rule="rewrite" parents="8179 1"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="8366">
    <literal><![CDATA[
      f(x,f(c_0,f(y,f(y,x)))) = c_0
    ]]></literal>
    <justification jstring="[back_rewrite(8128),rewrite(8179(4),1(4),8179(7))].">
      <j1 rule="back_rewrite" parents="8128"/>
      <j2 rule="rewrite" parents="8179 1 8179"/>
    </justification>
  </clause>

  <clause id="8419">
    <literal><![CDATA[
      f(c_0,f(c_0,x)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(8071),rewrite(8179(2),8179(3))].">
      <j1 rule="back_rewrite" parents="8071"/>
      <j2 rule="rewrite" parents="8179 8179"/>
    </justification>
  </clause>

  <clause id="8423">
    <literal><![CDATA[
      f(c_0,f(x,x)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(8065),rewrite(8179(2))].">
      <j1 rule="back_rewrite" parents="8065"/>
      <j2 rule="rewrite" parents="8179"/>
    </justification>
  </clause>

  <clause id="8425">
    <literal><![CDATA[
      f(c_0,f(x,c_0)) = x
    ]]></literal>
    <justification jstring="[back_rewrite(8063),rewrite(8179(2),8179(3))].">
      <j1 rule="back_rewrite" parents="8063"/>
      <j2 rule="rewrite" parents="8179 8179"/>
    </justification>
  </clause>

  <clause id="8426">
    <literal><![CDATA[
      f(x,f(y,c_0)) = f(x,f(y,x))
    ]]></literal>
    <justification jstring="[back_rewrite(8061),rewrite(8179(2))].">
      <j1 rule="back_rewrite" parents="8061"/>
      <j2 rule="rewrite" parents="8179"/>
    </justification>
  </clause>

  <clause id="8430">
    <literal><![CDATA[
      f(x,f(c_0,f(y,x))) = f(y,x)
    ]]></literal>
    <justification jstring="[back_rewrite(8056),rewrite(8179(3),1(3))].">
      <j1 rule="back_rewrite" parents="8056"/>
      <j2 rule="rewrite" parents="8179 1"/>
    </justification>
  </clause>

  <clause id="8446">
    <literal><![CDATA[
      f(c_0,x) = f(x,x)
    ]]></literal>
    <justification jstring="[back_rewrite(8021),rewrite(8179(2))].">
      <j1 rule="back_rewrite" parents="8021"/>
      <j2 rule="rewrite" parents="8179"/>
    </justification>
  </clause>

  <clause id="8859">
    <literal><![CDATA[
      f(f(x,y),f(z,c_0)) = f(f(z,z),f(x,y))
    ]]></literal>
    <justification jstring="[back_rewrite(784),rewrite(8426(4,R))].">
      <j1 rule="back_rewrite" parents="784"/>
      <j2 rule="rewrite" parents="8426"/>
    </justification>
  </clause>

  <clause id="9473">
    <literal><![CDATA[
      f(f(x,y),f(c_0,y)) = y
    ]]></literal>
    <justification jstring="[para(8179(a,1),12(a,1,2,1))].">
      <j1 rule="para" parents="8179 12"/>
    </justification>
  </clause>

  <clause id="9541">
    <literal><![CDATA[
      f(f(c_0,f(x,y)),f(f(c_0,y),z)) = f(x,y)
    ]]></literal>
    <justification jstring="[para(23(a,1),8419(a,1,2)),rewrite(8419(5)),flip(a)].">
      <j1 rule="para" parents="23 8419"/>
      <j2 rule="rewrite" parents="8419"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="9542">
    <literal><![CDATA[
      f(f(x,c_0),f(y,f(c_0,f(x,z)))) = x
    ]]></literal>
    <justification jstring="[para(24(a,1),8419(a,1,2)),rewrite(8425(4)),flip(a)].">
      <j1 rule="para" parents="24 8419"/>
      <j2 rule="rewrite" parents="8425"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="9631">
    <literal><![CDATA[
      f(x,f(c_0,y)) = f(x,f(y,x))
    ]]></literal>
    <justification jstring="[para(8446(a,2),569(a,1,2))].">
      <j1 rule="para" parents="8446 569"/>
    </justification>
  </clause>

  <clause id="9634">
    <literal><![CDATA[
      f(x,f(c_0,y)) = f(x,f(x,y))
    ]]></literal>
    <justification jstring="[para(8446(a,2),588(a,1,2))].">
      <j1 rule="para" parents="8446 588"/>
    </justification>
  </clause>

  <clause id="9637">
    <literal><![CDATA[
      f(x,f(x,f(c_0,y))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(8446(a,2),825(a,1,2,2))].">
      <j1 rule="para" parents="8446 825"/>
    </justification>
  </clause>

  <clause id="9645">
    <literal><![CDATA[
      f(f(c_0,x),y) = f(y,f(x,x))
    ]]></literal>
    <justification jstring="[para(8446(a,2),90(a,2,1)),rewrite(44(3)),flip(a)].">
      <j1 rule="para" parents="8446 90"/>
      <j2 rule="rewrite" parents="44"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="10194">
    <literal><![CDATA[
      f(x,f(y,f(x,f(z,f(y,x))))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(354(a,1),67(a,1,2)),flip(a)].">
      <j1 rule="para" parents="354 67"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="10349">
    <literal><![CDATA[
      f(f(x,y),f(c_0,f(x,f(c_0,f(y,z))))) = c_0
    ]]></literal>
    <justification jstring="[para(16(a,1),8366(a,1,2,2,2)),rewrite(1(5),9631(5,R))].">
      <j1 rule="para" parents="16 8366"/>
      <j2 rule="rewrite" parents="1 9631"/>
    </justification>
  </clause>

  <clause id="12119">
    <literal><![CDATA[
      f(f(x,f(c_0,f(y,z))),f(y,c_0)) = y
    ]]></literal>
    <justification jstring="[para(24(a,1),8430(a,1,2)),rewrite(9542(14))].">
      <j1 rule="para" parents="24 8430"/>
      <j2 rule="rewrite" parents="9542"/>
    </justification>
  </clause>

  <clause id="12148">
    <literal><![CDATA[
      f(f(c_0,f(x,y)),f(z,f(c_0,x))) = f(x,y)
    ]]></literal>
    <justification jstring="[para(72(a,1),8419(a,1,2)),rewrite(8419(5)),flip(a)].">
      <j1 rule="para" parents="72 8419"/>
      <j2 rule="rewrite" parents="8419"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="12350">
    <literal><![CDATA[
      f(f(x,c_0),f(y,f(c_0,f(z,x)))) = x
    ]]></literal>
    <justification jstring="[para(75(a,1),8419(a,1,2)),rewrite(8425(4)),flip(a)].">
      <j1 rule="para" parents="75 8419"/>
      <j2 rule="rewrite" parents="8425"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="12372">
    <literal><![CDATA[
      f(f(x,f(c_0,f(y,z))),f(z,c_0)) = z
    ]]></literal>
    <justification jstring="[para(75(a,1),8430(a,1,2)),rewrite(12350(14))].">
      <j1 rule="para" parents="75 8430"/>
      <j2 rule="rewrite" parents="12350"/>
    </justification>
  </clause>

  <clause id="13907">
    <literal><![CDATA[
      f(f(x,f(x,y)),f(x,f(y,z))) = f(c_0,f(x,f(y,z)))
    ]]></literal>
    <justification jstring="[para(86(a,1),106(a,1,2)),rewrite(8446(5,R),1(6)),flip(a)].">
      <j1 rule="para" parents="86 106"/>
      <j2 rule="rewrite" parents="8446 1"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="14572">
    <literal><![CDATA[
      f(f(f(x,y),z),f(f(u,f(y,y)),z)) = z
    ]]></literal>
    <justification jstring="[para(63(a,1),644(a,1,1,1)),rewrite(1(8),168(8))].">
      <j1 rule="para" parents="63 644"/>
      <j2 rule="rewrite" parents="1 168"/>
    </justification>
  </clause>

  <clause id="14926">
    <literal><![CDATA[
      f(f(f(f(x,x),y),z),f(f(u,x),z)) = z
    ]]></literal>
    <justification jstring="[para(46(a,1),915(a,1,1,1)),rewrite(8446(8,R),8423(8),1(8),64(8))].">
      <j1 rule="para" parents="46 915"/>
      <j2 rule="rewrite" parents="8446 8423 1 64"/>
    </justification>
  </clause>

  <clause id="15532">
    <literal><![CDATA[
      f(x,f(y,f(f(y,x),z))) = f(x,f(y,y))
    ]]></literal>
    <justification jstring="[para(1950(a,1),32(a,1,2)),flip(a)].">
      <j1 rule="para" parents="1950 32"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="16058">
    <literal><![CDATA[
      f(x,f(y,f(f(x,y),z))) = f(x,f(y,y))
    ]]></literal>
    <justification jstring="[para(5676(a,1),32(a,1,2)),flip(a)].">
      <j1 rule="para" parents="5676 32"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="16257">
    <literal><![CDATA[
      f(x,f(y,f(f(x,x),z))) = f(x,f(y,y))
    ]]></literal>
    <justification jstring="[para(7921(a,1),32(a,1,2)),flip(a)].">
      <j1 rule="para" parents="7921 32"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="16272">
    <literal><![CDATA[
      f(x,f(y,f(c_0,f(f(y,x),z)))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(7921(a,1),96(a,1)),rewrite(8446(6,R),1(6)),flip(a)].">
      <j1 rule="para" parents="7921 96"/>
      <j2 rule="rewrite" parents="8446 1"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="16879">
    <literal><![CDATA[
      f(f(x,y),f(z,c_0)) = f(f(z,z),f(y,x))
    ]]></literal>
    <justification jstring="[para(1(a,1),8859(a,1,1))].">
      <j1 rule="para" parents="1 8859"/>
    </justification>
  </clause>

  <clause id="17270">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(f(f(c_0,z),u),x)) = x
    ]]></literal>
    <justification jstring="[para(9541(a,1),31(a,1,1,2))].">
      <j1 rule="para" parents="9541 31"/>
    </justification>
  </clause>

  <clause id="17895">
    <literal><![CDATA[
      f(x,f(y,f(z,f(x,y)))) = f(x,f(y,y))
    ]]></literal>
    <justification jstring="[para(10194(a,1),32(a,1,2)),flip(a)].">
      <j1 rule="para" parents="10194 32"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="17940">
    <literal><![CDATA[
      f(x,f(y,f(c_0,f(z,f(y,x))))) = f(x,x)
    ]]></literal>
    <justification jstring="[para(142(a,1),7921(a,1)),rewrite(8446(5,R),1(5))].">
      <j1 rule="para" parents="142 7921"/>
      <j2 rule="rewrite" parents="8446 1"/>
    </justification>
  </clause>

  <clause id="18864">
    <literal><![CDATA[
      f(f(x,x),f(c_0,f(f(y,x),z))) = f(f(x,x),z)
    ]]></literal>
    <justification jstring="[para(14572(a,1),15532(a,1,2)),rewrite(8446(8,R)),flip(a)].">
      <j1 rule="para" parents="14572 15532"/>
      <j2 rule="rewrite" parents="8446"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="18866">
    <literal><![CDATA[
      f(x,f(c_0,f(f(f(x,x),y),z))) = f(x,z)
    ]]></literal>
    <justification jstring="[para(14926(a,1),15532(a,1,2)),rewrite(8446(8,R)),flip(a)].">
      <j1 rule="para" parents="14926 15532"/>
      <j2 rule="rewrite" parents="8446"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="18903">
    <literal><![CDATA[
      f(x,f(y,f(f(x,x),z))) = f(x,f(y,x))
    ]]></literal>
    <justification jstring="[para(16257(a,2),569(a,1))].">
      <j1 rule="para" parents="16257 569"/>
    </justification>
  </clause>

  <clause id="19058">
    <literal><![CDATA[
      f(x,f(c_0,f(y,z))) = f(x,f(x,f(z,y)))
    ]]></literal>
    <justification jstring="[para(16879(a,2),288(a,2,2)),rewrite(8446(3,R),8423(3),18903(7),1(4),9(4),1(7),8423(7),9631(6,R)),flip(a)].">
      <j1 rule="para" parents="16879 288"/>
      <j2 rule="rewrite" parents="8446 8423 18903 1 9 1 8423 9631"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="19301">
    <literal><![CDATA[
      f(f(c_0,x),f(c_0,f(y,f(z,x)))) = f(f(c_0,x),y)
    ]]></literal>
    <justification jstring="[para(17270(a,1),16058(a,1,2)),rewrite(8446(10,R)),flip(a)].">
      <j1 rule="para" parents="17270 16058"/>
      <j2 rule="rewrite" parents="8446"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="19334">
    <literal><![CDATA[
      f(x,f(f(y,y),z)) = f(x,f(f(x,y),z))
    ]]></literal>
    <justification jstring="[para(16272(a,1),17895(a,1,2,2)),rewrite(1(6),18864(6),8446(12,R),8419(9))].">
      <j1 rule="para" parents="16272 17895"/>
      <j2 rule="rewrite" parents="1 18864 8446 8419"/>
    </justification>
  </clause>

  <clause id="19348">
    <literal><![CDATA[
      f(x,f(f(c_0,y),z)) = f(x,f(z,f(x,y)))
    ]]></literal>
    <justification jstring="[para(17940(a,1),17895(a,1,2,2)),rewrite(9645(6,R),19301(7),8446(13,R),8419(10))].">
      <j1 rule="para" parents="17940 17895"/>
      <j2 rule="rewrite" parents="9645 19301 8446 8419"/>
    </justification>
  </clause>

  <clause id="20289">
    <literal><![CDATA[
      f(f(c_0,f(x,y)),f(c_0,f(x,z))) = f(f(c_0,f(x,y)),z)
    ]]></literal>
    <justification jstring="[para(12119(a,1),18866(a,1,2,2,1))].">
      <j1 rule="para" parents="12119 18866"/>
    </justification>
  </clause>

  <clause id="20290">
    <literal><![CDATA[
      f(f(c_0,f(x,y)),f(c_0,f(y,z))) = f(f(c_0,f(x,y)),z)
    ]]></literal>
    <justification jstring="[para(12372(a,1),18866(a,1,2,2,1))].">
      <j1 rule="para" parents="12372 18866"/>
    </justification>
  </clause>

  <clause id="20504">
    <literal><![CDATA[
      f(f(c_0,x),f(y,f(x,z))) = f(f(c_0,x),f(y,c_0))
    ]]></literal>
    <justification jstring="[para(9473(a,1),18903(a,1,2,2,1)),rewrite(8426(11,R))].">
      <j1 rule="para" parents="9473 18903"/>
      <j2 rule="rewrite" parents="8426"/>
    </justification>
  </clause>

  <clause id="20508">
    <literal><![CDATA[
      f(f(c_0,f(x,y)),z) = f(x,f(c_0,f(y,z)))
    ]]></literal>
    <justification jstring="[para(10349(a,1),18903(a,2,2)),rewrite(8446(20,R),8419(15),20504(14),1(9),1(10),20289(10),20290(7),1(12),8419(12))].">
      <j1 rule="para" parents="10349 18903"/>
      <j2 rule="rewrite" parents="8446 8419 20504 1 1 20289 20290 1 8419"/>
    </justification>
  </clause>

  <clause id="20512">
    <literal><![CDATA[
      f(x,f(y,f(c_0,f(z,x)))) = f(x,f(y,z))
    ]]></literal>
    <justification jstring="[para(12148(a,1),18903(a,1,2)),rewrite(20508(6)),flip(a)].">
      <j1 rule="para" parents="12148 18903"/>
      <j2 rule="rewrite" parents="20508"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="23926">
    <literal><![CDATA[
      f(f(x,f(x,y)),f(x,z)) = f(c_0,f(x,f(y,f(x,z))))
    ]]></literal>
    <justification jstring="[para(19334(a,2),295(a,1,2)),rewrite(8446(5,R),20508(7),9637(8),13907(6)),flip(a)].">
      <j1 rule="para" parents="19334 295"/>
      <j2 rule="rewrite" parents="8446 20508 9637 13907"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="24110">
    <literal><![CDATA[
      f(c_0,f(a,f(b,f(a,f(a,c))))) != f(c_0,f(a,f(b,c)))
    ]]></literal>
    <attribute><![CDATA[
      answer("Sheffer")
    ]]></attribute>
    <justification jstring="[back_rewrite(8325),rewrite(23926(11))].">
      <j1 rule="back_rewrite" parents="8325"/>
      <j2 rule="rewrite" parents="23926"/>
    </justification>
  </clause>

  <clause id="24343">
    <literal><![CDATA[
      f(c_0,f(a,f(b,f(b,f(a,c))))) != f(c_0,f(a,f(b,c)))
    ]]></literal>
    <attribute><![CDATA[
      answer("Sheffer")
    ]]></attribute>
    <justification jstring="[para(19348(a,2),24110(a,1,2)),rewrite(1(9),9634(9))].">
      <j1 rule="para" parents="19348 24110"/>
      <j2 rule="rewrite" parents="1 9634"/>
    </justification>
  </clause>

  <clause id="24498">
    <literal><![CDATA[
      f(x,f(y,f(y,f(x,z)))) = f(x,f(y,z))
    ]]></literal>
    <justification jstring="[para(19058(a,1),20512(a,1,2))].">
      <j1 rule="para" parents="19058 20512"/>
    </justification>
  </clause>

  <clause id="24500">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer("Sheffer")
    ]]></attribute>
    <justification jstring="[back_rewrite(24343),rewrite(24498(10)),xx(a)].">
      <j1 rule="back_rewrite" parents="24343"/>
      <j2 rule="rewrite" parents="24498"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
